Full text unavailable from EThOS. Please contact the current institution’s library for further details.

Abstract:

The underlying motivation of the thesis is to generalise the techniques of Buzzard-Taylor and Buzzard to a totally real field F. Their novel approach to modularity of Galois representations via knowing conditions under which an overconvergent Up-eigenform is indeed a classical modular forms, was instrumental in proving new cases of Arlin conjecture for Q and has also been important in the modern theory of p-adic modular forms. My thesis is an exploration of their ideas in the Hilbert case.
Analytic continuation of overconvergent Hilbert eigenfor~s depends fundamentally on the rigid geometry of Hilbert modular varieties. The model Deligne-Pappas considered, is highly singular at fibres over ramified primes and in order to avoid technical problems arising from geometry, I construct, in the first chapter, a model
over the integers of a finite extension of Qp which desingularises the Deligne-Pappas model, using ideas from Pappas-Rapoport on local models.
In the second chapter, I prove an analogue. in the Hilbert case where p splits completely in F, of Coleman's theorem- an overconvergent Up-eigenform of small slope is classical. The methodology is to generalise the
earlier work of Buzzard on analytic continuation of overconvergent Up-eigenforms (non-ordinary loci) and then apply Kassaei's idea (ordinary loci) to glue overconvergent forms in the absence of companion forms.